| L(s) = 1 | + 0.428·2-s + 3·3-s − 7.81·4-s + 1.28·6-s − 7·7-s − 6.77·8-s + 9·9-s − 27.4·11-s − 23.4·12-s + 46.5·13-s − 2.99·14-s + 59.6·16-s − 5.20·17-s + 3.85·18-s − 91.0·19-s − 21·21-s − 11.7·22-s + 111.·23-s − 20.3·24-s + 19.9·26-s + 27·27-s + 54.7·28-s + 0.0763·29-s + 201.·31-s + 79.7·32-s − 82.4·33-s − 2.22·34-s + ⋯ |
| L(s) = 1 | + 0.151·2-s + 0.577·3-s − 0.977·4-s + 0.0874·6-s − 0.377·7-s − 0.299·8-s + 0.333·9-s − 0.753·11-s − 0.564·12-s + 0.993·13-s − 0.0572·14-s + 0.931·16-s − 0.0742·17-s + 0.0504·18-s − 1.09·19-s − 0.218·21-s − 0.114·22-s + 1.01·23-s − 0.172·24-s + 0.150·26-s + 0.192·27-s + 0.369·28-s + 0.000488·29-s + 1.16·31-s + 0.440·32-s − 0.434·33-s − 0.0112·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.826750627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.826750627\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 0.428T + 8T^{2} \) |
| 11 | \( 1 + 27.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.20T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 0.0763T + 2.43e4T^{2} \) |
| 31 | \( 1 - 201.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 312.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 257.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 350.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 196.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 881.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 737.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 365.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 273.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 87.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 228.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.28428977415206813629588537676, −9.496702604534526226520313830199, −8.579328295815629880717146469295, −8.113889700924215525233943467684, −6.78245618019486163805759016474, −5.72241263750348791563088547170, −4.60327916507291135717766766167, −3.70449572061600321806485833114, −2.59429285907293877386833340575, −0.802185968465924028207584599825,
0.802185968465924028207584599825, 2.59429285907293877386833340575, 3.70449572061600321806485833114, 4.60327916507291135717766766167, 5.72241263750348791563088547170, 6.78245618019486163805759016474, 8.113889700924215525233943467684, 8.579328295815629880717146469295, 9.496702604534526226520313830199, 10.28428977415206813629588537676
